Literature review.   Mixed-monotone decomposition functions are generally  not unique, hence several seminal studies have addressed the issue of identifying and/or constructing appropriate decomposition functions with somewhat different yet highly related definitions and corresponding sufficient conditions for mixed-monotonicity [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. In particular,
recent studies in [1]}, [3]} provided tight decomposition functions for unconstrained discrete -time and continuous-time dynamical systems, respectively,  whose computability rely on  the global solvability of nonlinear optimization programs, which is  only guaranteed in  some specific cases such as when the vector field is Jacobian sign-stable, or when all  its  stationary/critical
points can be  computed analytically. On the other hand,  an interesting study in [2]} proposed computable and constructive (but not necessarily tight) decomposition functions for differentiable vector fields with known bounds for the derivatives.  Building on these frameworks, we aim to obtain computable/tractable and tighter decomposition functions for a broad class of nonsmooth systems.
